Finite Group Databasebeta

← #227013

Subgroups — #227013

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23095C211yesmaximal, characteristic, complemented
2#22995C211yesmaximal, characteristic
3#22519C211yesmaximal, characteristic, complemented
4#22487C211yesmaximal, characteristic, complemented
5C2×#1622C211yesmaximal, characteristic
6#9820C211yesmaximal, characteristic, complemented
7#9568C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9222C211yesmaximal, characteristic, complemented
10#9221C211yesmaximal, characteristic
11#9064C211yesmaximal, characteristic, complemented
12#8887C211yesmaximal, characteristic, complemented
13#8886C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic, complemented
16C2×#554C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#459C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C4×D8—21no—
22C2×C8×D4C2211yescharacteristic
23#2366C2211yescharacteristic
24#2355C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#1991—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1989—21no—
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1968—21no—
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1934—21no—
38#1926—21no—
39#1892—21no—
40#1879—21no—
41#1877—21no—
42#1872—21no—
43#1862—21no—
44#1844C2211yescharacteristic
45#1824—21no—
46#1804—21no—
47#1803C2211yescharacteristic
48#1802C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C4×#103C2211yescharacteristic
52C4×#102—21no—
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56#1622C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—41no—
70C22×D8—21no—
71C23×C8—21no—
72#554—22no—
73#554—22no—
74#554—41no—
75#554C2×C414yescomplemented
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#125—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—41no—
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89#492—41no—
90#492C2×C414yescomplemented
91#460—22no—
92#460C2×C414yescomplemented
93#459—22no—
94#459C2×C414yes—
95C8×Q8—22no—
96C8×Q8—41no—
97C4×D8—44no—
98C8×D4—22no—
99C8×D4—41no—
100C2×#107—21no—
101C2×#106—21no—
102C2×#106—41no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103—21no—
109C2×#103D411yescharacteristic
110C2×#103D411yescharacteristic
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102D411yescharacteristic
119C2×#102D411yescharacteristic
120C2×C22⋊C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391—21no—
128#390—21no—
129#390—21no—
130#389—21no—
131#387—41no—
132#386—41no—
133#385—41no—
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#382—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#336—21no—
159#336—21no—
160#333C2×C414yescomplemented
161#331—41no—
162Q8⋊C8C2×C414yescomplemented
163D4⋊C8C2×C414yescomplemented
164C82—41no—
165C22×Q8—21no—
166C22×Q8C2×D411yescharacteristic
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C2×D411yescharacteristic
171C23×C4—21no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C2×SD16—24no—
176C2×SD16—44no—
177C2×SD16—82no—
178C2×SD16—82no—
179C2×SD16—81no—
180C2×SD16—81no—
181C2×D8—24no—
182C2×D8—42no—
183C2×D8—42no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—81no—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8C22×C412yes—
200#127—21no—
201#127—21no—
202#127—21no—
203#127C22×C412yes—
204#125—24no—
205#125—42no—
206C22⋊Q8—24no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209C22⋊Q8—42no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8C22×C412yes—
213C4×Q8#4112yes—
214C4×D4—24no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—81no—
222C4×D4—81no—
223C4×D4—81no—
224C4×D4C22×C412yes—
225C4×D4#4112yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—81no—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—41no—
258C2×C42C2411yescharacteristic
259#107—42no—
260#106—81no—
261#106—82no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—42no—
269C4⋊C8C22×C412yes—
270C4⋊C8C22×C412yes—
271#103—22no—
272#103—22no—
273#103—81no—
274#102—22no—
275#102—24no—
276#102—24no—
277#102—42no—
278#102—42no—
279#102—42no—
280#102—81no—
281#102—81no—
282#102SD1614yescomplemented
283C22⋊C8—42no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—42no—
290C4×C8—42no—
291C4×C8—81no—
292C4×C8C22×C412yes—
293#95—21no—
294#95—21no—
295#95—41no—
296#95—41no—
297#95—41no—
298#95—41no—
299C24—21no—
300C24—41no—
301C2×Q8—21no—
302C2×Q8—22no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—41no—
308C2×Q8—81no—
309C2×Q8—81no—
310C2×Q8C4×D412yes—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—81no—
329C2×D4—81no—
330C2×D4C4×D412yes—
331C2×D4C2×SD1612yes—
332C2×D4#13512yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—161no—
355C22×C4—161no—
356C22×C4—161no—
357C22×C4—161no—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361SD16—88no—
362SD16—84no—
363D8—48no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—84no—
390C2×C8C2×SD1612yes—
391C2×C8#13512yes—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—82no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4C4×D412yes—
420C4⋊C4C4×D412yes—
421C4⋊C4C2×SD1612yes—
422C4⋊C4#13512yes—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—161no—
430C42—22no—
431C42—22no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—161no—
437C42C23×C411yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4111yescharacteristic
440C42H(5,𝔽2)11yescharacteristic
441C23—22no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23D4211yescharacteristic
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—82no—
461Q8—161no—
462D4—22no—
463D4—22no—
464D4—22no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—82no—
470D4—81no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—161no—
475D4C4×SD1614yes—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—164no—
513C2×C4—164no—
514C2×C4—164no—
515C2×C4C2×C4×D411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4#51811yescharacteristic
518C2×C4#54611yescharacteristic
519C2×C4#54611yescharacteristic
520C2×C4#55111yescharacteristic
521C2×C4C22×SD1611yescharacteristic
522C2×C4C2×#13511yescharacteristic
523C2×C4C2×#13511yescharacteristic
524C2×C4C2×#13511yescharacteristic
525C2×C4#57611yescharacteristic
526C2×C4#57711yescharacteristic
527C2×C4#57711yescharacteristic
528C2×C4#57811yescharacteristic
529C8—24no—
530C8—22no—
531C8—22no—
532C8—42no—
533C8—42no—
534C8—41no—
535C8—41no—
536C8—81no—
537C8—81no—
538C22—24no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—161no—
549C22—161no—
550C22—161no—
551C22—161no—
552C22#235811yescharacteristic
553C22#328011yescharacteristic
554C22D4×SD1611yescharacteristic
555C22#328411yescharacteristic
556C22#328511yescharacteristic
557C22#329111yescharacteristic
558C22#329211yescharacteristic
559C4—21no—
560C4—21no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—82no—
569C4—161no—
570C4—161no—
571C4C2×C4×SD1611yescharacteristic
572C4C4×#13511yescharacteristic
573C4#294611yescharacteristic
574C4#294711yescharacteristic
575C4#329711yescharacteristic
576C4#329811yescharacteristic
577C4#330111yescharacteristic
578C4#330311yescharacteristic
579C2—44no—
580C2—162no—
581C2#1997611yescharacteristic
582C2#1998211yescharacteristic
583C2#1999211yescharacteristic
584C2#2001211yescharacteristic
585C2#2287211yescharacteristic
586C2#2287511yescharacteristic
587C2#2289711yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23095C2semidirect
3. #22519C2semidirect
4. #22487C2semidirect
6. #9820C2semidirect
7. #9568C2semidirect
8. #9567C2semidirect
9. #9222C2semidirect
11. #9064C2semidirect
12. #8887C2semidirect
13. #8886C2semidirect
14. C8×#103C2semidirect
15. C8×#102C2semidirect
56. #1622C4semidirect
75. #554C2×C4semidirect
90. #492C2×C4semidirect
92. #460C2×C4semidirect
160. #333C2×C4semidirect
162. Q8⋊C8C2×C4semidirect
163. D4⋊C8C2×C4semidirect
282. #102SD16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#230952
#229952
#225192
#224872
C2×#16222
#98202
#95682
#95672
#92222
#92212
#90642
#88872
#88862
C8×#1032
C8×#1022